To solve this problem, we need to determine whether each given function is even, odd, or neither. Here’s how we can assess each one based on symmetry properties.

1. Function $r(x)$:

   • The graph of $r(x)$ is symmetric with respect to the $y$-axis, indicating that it is an even function.
   • A function is considered even if $f(x) = f(-x)$ for all $x$.

2. Function $s(x)$:

   • The graph of $s(x)$ appears to have rotational symmetry about the origin, indicating it could be an odd function.
   • A function is considered odd if $f(x) = -f(-x)$ for all $x$.

3. Function $g(x) = -2x^4 + 5x^2$:

   • Analyzing the terms:
     • $x^4$ and $x^2$ are even powers, and multiplying by a constant doesn’t affect this.
     • Both terms $x^4$ and $x^2$ would produce even function behavior because replacing $x$ with $-x$ results in the same expression.
   • Therefore, $g(x)$ is an even function.

4. Function $h(x) = 2x^5$:

   • The power of $x$ is odd (5), and so this function behaves as an odd function.
   • Replacing $x$ with $-x$ results in $h(-x) = -2x^5 = -h(x)$, confirming that it is an odd function.

Summary:

• Function $r$: Even
• Function $s$: Odd
• Function $g(x) = -2x^4 + 5x^2$: Even
• Function $h(x) = 2x^5$: Odd

Would you like more detailed explanations or have any further questions?

Related Questions:

1. What are the specific properties of even and odd functions in terms of symmetry?
2. How do the signs and exponents of terms in a polynomial affect whether a function is even, odd, or neither?
3. Can a function be both even and odd? If so, under what conditions?
4. What is the importance of identifying symmetry in functions in calculus or physics?
5. How would transformations (like shifts or stretches) affect the symmetry of a function?

Tip:

To quickly determine if a function is even or odd, try substituting $-x$ for $x$.